Counting atoms: corners, faces and centers are shared
Each corner atom is split across 8 neighboring cells, and each face atom across 2. So the 8 corners of any cube contribute just 1 atom. FCC adds 6 face atoms (6 × ½ = 3) for 4 total; BCC adds one whole atom at the center for 2 total.
The three cubic structures at a glance
| Structure | Atoms/cell | Coordination no. | Packing (APF) | a vs radius R |
|---|---|---|---|---|
| Simple cubic (SC) | 1 | 6 | 52.4% | a = 2R |
| Body-centered (BCC) | 2 | 8 | 68.0% | a = 4R/√3 |
| Face-centered (FCC) | 4 | 12 | 74.0% | a = 4R/√2 |
Iron, chromium and tungsten are BCC. Copper, aluminum, silver and gold are FCC. Simple cubic is rare in pure metals.
Why FCC packs so much tighter than simple cubic
The packing factor is the same idea for all three: how much of the box is actually solid atom rather than empty space. FCC squeezes in 40% more atom by volume than simple cubic.
FCC is the tightest arrangement of equal spheres there is. Denser packing and 12 nearest neighbors are why FCC metals like copper are so ductile.
Getting crystal density from the cell
Z is atoms per cell, M the molar mass, a the edge length, and NA Avogadro's number. For copper (FCC, Z = 4, M = 63.55 g/mol, a = 3.615 Å) this gives about 8.96 g/cm³, which matches the measured value.
Common questions
How many atoms are in a cubic unit cell?
It depends on the type. Simple cubic has 1 atom, body-centered cubic (BCC) has 2, and face-centered cubic (FCC) has 4. A corner atom counts as one-eighth, a face atom as one-half, and a center atom as a whole.
What is the atomic packing factor?
It is the fraction of the cell volume filled by atoms. Simple cubic is about 52 percent, BCC about 68 percent, and FCC about 74 percent. FCC is the densest packing possible for equal spheres, tied with hexagonal close packing.
How do I get the lattice parameter from the atomic radius?
The relation differs by structure because the atoms touch along different lines. Simple cubic: a equals 2R. BCC: a equals 4R divided by the square root of 3. FCC: a equals 4R divided by the square root of 2.
